Is Nature Accessible to the Mathematical Physicist?

by
William A. Wallace, O.P.

The topic I wish to explore in this paper should be of special
interest to Thomists, for St. Thomas's teachings on the relationships
between physical science and mathematics are distinctive and relevant
to problems faced by scientists and philosophers in the present day.
Yet there are difficulties in addressing this topic from the side of
both physics and mathematics, since these disciplines bear little
resemblance to those that went under the same names in the thirteenth
century. Again, natural philosophy in the Thomistic tradition is in a
relatively undeveloped state, having suffered from an overdose of
metaphysics, even from metaphysical imperialism, in the past half
century. So I am forced to build on foundations I myself have laid in
my The Modeling of Nature, where I developed a philosophy of
science based on an Aristotelian-Thomistic philosophy of nature, and in
the paper I read at this Institute last year, where I elaborated on two
subjects not treated in that book, the dispositions of protomatter and
the use of mathematical models in the study of nature.1

The following is an overview of the thesis I intend to develop.
Simply put, it answers the question I propose in my title in the
affirmative. Yes, nature is accessible to the mathematical physicist.
This can be seen readily by anyone who accepts mathematical physics as
a scientia media or "mixed science" in the Aristotelian-Thomistic
sense. Such a science enables one to demonstrate properties
of natural bodies and to grasp the natures of inorganic substances, the
elements and compounds that make up the universe in which we live.
Also, mathematical physics enables us to attain scientific knowledge of
physical bodies composed of elements and compounds, such as stars and
planets, to the extent that they have natures, and even of subatomic
entities that enter into their composition. In the case of subatomic
entities the qualification "to the extent that they have natures" opens
up the problem of what I call "transient natures," which I shall
address in the latter part of this paper.2

The primary instrument of a mathematical physics is a demonstrative
syllogism composed of two premises, one physical or natural, the other
mathematical. The middle term will generally be a metrical concept,
that is, one that expresses the result of a measuring process that
applies a number or figure to a physical entity, and so pertains to
both mathematics and physics. Such a measurement, even though
approximate, is regarded as true if its result falls within the limits
of error of the measuring process. The middle term may also take on
meanings that are partly the same and partly different in the two
premises, thus making use of analogous predication. Again, suppositions
may be needed for the demonstration, and these require previous assent
or verification independently of their use in the demonstration. These
qualifications understood, it is possible for the mathematical
physicist to secure strict demonstrations and thus to possess true
scientific knowledge in the Aristotelian sense.

The Aristotelian Concept of Nature

Aristotle defines nature as "a principle and cause of being moved or
of rest in the thing to which it belongs primarily and in virtue of
that thing, but not accidentally" (Physics II.1, 192b21-23).
He further identifies it with the two essential principles of natural
things he earlier uncovered in the Physics, matter or
"underlying nature" (191a8), which I call protomatter, and a "natural
form" (192b1) that makes the thing be the kind it is. Aristotle
uncovers these principles from an analysis of the way in which
substances come to be and pass away in the order of nature. His
analysis is based not on mathematics but on ordinary experience,
through a study of the qualities of objects as presented to the senses.
Protomatter is for him a purely potential and indeterminate principle,
whereas natural form is an actualizing or determining principle, one
that specifies the object to be a particular natural kind.

In the order of the non-living, following an earlier Greek tradition
Aristotle recognized four elements as natural kinds: fire, air, water,
and earth. These elements he differentiated from one another by
combinations of primary qualities (hot and cold, wet and dry) and
motive powers (gravity and levity in different degrees). From them as
constituents he attempted to explain compounds or "mixed" bodies and
their various secondary qualities, such as their shapes and surface
characteristics.

The coming-to-be of a natural substance, for Aristotle, could be
brought about by the alteration of its sensible qualities. It was in
this way that his commentators came to explain the transmutation of the
elements, that is, the natural change of one element into
another.3 A schema called the symbolum was commonly
used to detail how this came about. This is shown in Fig. 1. Here the
two pairs of contraries occupy the two diameters of a circle, with
protomatter at its center. The contraries are arranged in such a way
that any two pairs can be regarded as having a quality in common with,
as well as a quality different from, that on either side of it. Thus
the hot-dry combination at the top is so related to the cold-dry on the
left that the hot-cold are extremes and dry-dry are common
intermediates. Similarly, the hot-dry is so related to the hot-wet on
the right that dry-wet are extremes and hot-hot are common
intermediates. The corresponding elements, each having its distinctive
pair of contraries, can be converted into one another through the
underlying substrate in the center, protomatter. Whenever an extrinsic
agent so affects the dispositions of protomatter as to cause one or
other distinctive pair of primary qualities to become dominant, the
element corresponding to that pair of contraries emerges naturally from
the potency of protomatter, the previous element recedes back into that
potency, and an elemental transformation has taken
place.4

For purposes of later reference, it will be convenient to replace
the symbolum with the matrix shown in Table I, where the
protomatter is not shown but nonetheless is presupposed.

Here the first column lists the elements (Fire, Air, Water, Earth),
the second, the presence of heat (1) or its absence cold (0) in the
element, and the third, the presence of moisture (1) or its absence (0)
in the same. Using this matrix and the binary digits it employs one can
characterize the four basic elements that exist above the level of
protomatter, that are convertible into one another by being reduced
back to protomatter, and then are educed from protomatter with a
different combination of properties.

How this is done is diagramed in Fig. 2, slightly different from
Fig. 1 but again presupposing protomatter in the background. Here
normal transitions occur around the perimeter of the figure when one
parameter is varied and the other remains unchanged, as in F to A, A to
W, W to E, and E to F. Diagonal transitions, those from F to W and E to
A, would generally not be allowed, because both parameters would have
to be varied at once, and no sensible quality would be conserved
throughout the change.

Three observations may be made about this Aristotelian analysis of
elementarity. First, the four elements have never actually been
observed in the universe, and in fact are unobservable because they
would require a pure admixture of primary qualities, which is never
experienced in bodies that come under sense observation. Yet the bridge
to the knowledge of these elements is qualitative knowledge. In other
words, it is only through the qualities that are known to exist in
macroscopic bodies that one is able to reason to the existence of
bodies endowed with idealized qualities that, in some way or other,
serve to explain the appearances of composed bodies.

Second, both the four elemental bodies and the substrate that is
their basic component, protomatter, may be said to be real, although
neither is real in the same way as an existent sensible body.
Protomatter is real only in the sense of being a potentiality for
assuming various natural forms, whereas the four elements are never
completely actual in any composite, but are always in some remiss state
corresponding to the various degrees of remission of their primary
qualities.

Third, qualities observed in the macroscopic domain are explained by
idealized qualities, and idealized qualities are explained by a
quality-less substrate. The substrate also lacks quantity, and thus is
radically unpicturable and unobservable. Yet it is knowable by
experience from a knowledge of substance and from an analysis of what
happens in substantial change. Thus both the elements and the substrate
serve as real explanatory principles of chemical transformation and of
the composition of bodies. They also meet the rather stringent
requirements of the logic of explanation. Heat is explained by non-heat,
quantity by non-quantity, etc., so there is no circularity in the
explanatory process.

"Hot-cold" and "wet-dry" may seem to be strange couplets with which
to start discussing elementary particles, but they are really not far
different from the "up-down" and "top-bottom" couplets used in recent
quark theory. To understand this we must move closer to the terminology
of modern science. We propose to do so first by sketching in broad
strokes the development of two mixed sciences known from antiquity,
mechanics and optics, and then the gradual replacement of qualitative
terms by their metrical equivalents in these disciplines, thus
permitting the use of mathematical equations to provide explanations
and predictions of extremely broad ranges of physical and chemical
phenomena.

Two Mixed Sciences: Mechanics and
Optics

Apart from the science whose proper subject matter is nature,
Aristotle recognized a number of mixed sciences, two of which
approximate the subject matter of modern physics, namely, mechanics and
optics. Mechanics is a science concerned with the forces and weights
involved in moving bodies, and also with the study of motion in its
quantitative aspects. Optics

is a science concerned with the study of light rays, and for
purposes here it is taken in a sense broad enough to include astronomy
as the study of light rays coming from heavenly bodies.

Both make extensive use of Euclidean geometry and of simple number
theory involving integers and various types of ratios and
proportions.

Mechanics took its beginning from the Mechanical Problems
once attributed to Aristotle but now known to have been composed after
his death by a member of his school. This was concerned with
mathematical analyses of the basic machines used for moving bodies,
such as the wheel and the lever, and with general principles these
analyses provide. A specialization within this discipline known as
statics (including hydrostatics) was developed extensively by
Archimedes in the third century BC, on the supposition that
mathematical figures can have weight and centers of gravity, and so can
arrive at positions of equilibrium. Another specialization known as
kinematics, which studies the spatio-temporal relationships involved in
the motion of bodies, was developed at Merton College in Oxford in the
fourteenth century. This is important for having introduced concepts
such as instantaneous velocity, which prepared the way for the
calculus. In the early seventeenth century Galileo Galilei experimented
with motion and formulated a restricted principle of inertia and laws
of falling bodies. Around the same time Johann Kepler arrived at his
three laws of planetary motion. By the end of the seventeenth century
Isaac Newton had synthesized all these results in his
Principia, the famous Mathematical Principles of Natural
Philosophy (1687), which supplied the basic definitions, laws, and
propositions on which most of classical mechanics is based.

Classical mechanics underwent further development throughout the
eighteenth and nineteenth centuries, mainly through advances in
mathematical formulation. Newton had used geometrical constructions
throughout the Principia, although the principles of the
calculus were behind his work. The problems he addressed were mainly
those of motion on earth and in the solar system, where the straight-line
motion of a mass point in empty space sufficed for his basic
paradigm. Later advances came from employing differential and integral
calculus to address problems in hydrodynamics, acoustics, and stress
formation in rigid and deformable bodies. Vibratory motions were
studied in detail, first in strings, then in membranes, then in
three-dimensional solids, and in these areas vector analysis and complex
number theory provided needed mathematical tools. Finally, in the early
twentieth century, Albert Einstein saw the importance of time in signal
transmission, and so introduced it as a fourth dimension in his new
relativistic mechanics. This he proposed in both special and general
theories, whose elaboration required additional mathematical techniques
for dealing with multi-dimensional spaces.5

Like mechanics, optics had its beginnings as a mixed science among
the Greeks, first with Euclid, who wrote an Optics concerned
with the direct transmission of light rays, then with Ptolemy, whose
Optics also treated their refraction by transparent fluids and
their reflection by mirrors of various shapes. Eudoxus, Apollonius, and
Hipparchus were mostly concerned with light rays coming from heavenly
bodies, and worked out advanced geometrical constructions for the
geocentric astronomy that was later formulated by Ptolemy in his
Almagest. The Middle Ages saw further developments in the
study of lenses, of radiant phenomena such as the rainbow, and theories
of vision. In the mid-seventeenth century Newton performed his famous
experimentum crucis, showing that sunlight is composed of
light rays of different colors with their own angles of refraction. He
professed not to know the nature of light, though he seems to have held
for a particulate theory. His views were improved on by Christian
Huygens, who showed how light was propagated through spherical
wavelets. Thomas Young advanced beyond this, proposing in the early
nineteenth century that light consists of vibrations that are
transverse to the direction of light's motion, with two possible modes
at right angles to each other.

The next breakthrough occurred in the mid-nineteenth century through
Michael Faraday's study of electricity and magnetism, which led
ultimately to the electromagnetic theory of light. This was put in
mathematical form by James Clerk Maxwell, who improved on Young's
suggestion and proposed in 1862 that "light consists in the transverse
undulations of the same medium which is the cause of electric and
magnetic phenomena." The equations he used to explain polarized light
and other poorly understood phenomena were very similar to the partial
differential equations and vector operators then being used in advanced
mechanics. Physical optics thus came to merge with analytical
mechanics. It likewise found ready application in Einstein's
relativistic mechanics when this developed in the early decades of the
twentieth century. More importantly still, Max Planck's study of
black-body radiation around 1900 led to the discovery that radiation is
emitted in very small packets of energy called quanta. This marked the
beginning of quantum mechanics -- yet a third "mixed science" destined
to be dominant throughout the twentieth century and one that poses
special problems of interest to scientists and philosophers alike.

Metrical Concepts: Quantifying
Qualities

Before turning to quantum mechanics, we must now consider a problem
that was touched on earlier but has yet to be addressed. The point has
been made that Aristotle's analysis in the Physics was based
on ordinary experience, not on mathematics, but rather on the study of
sensible qualities, that is, the qualities of bodies as presented to
the senses. Is there any way in which such qualities can be made
susceptible to mathematical treatment and thus enter into the reasoning
processes of a mixed science? Within Thomism it would seem that this
question can be answered in the affirmative. The basis for this reply
is St. Thomas's teaching that there is a hierarchical ordering among
the accidents found in a natural body, with quantity being the most
fundamental, and with the remaining accidents coming to substance
through quantity as an intermediate (quantitate mediante).
Being received into substance through quantity, sensible qualities have
a quantitative aspect, and it is on this basis that they can be
measured. Being measurable, they themselves can take on the formality
of metrical concepts, and so serve as middle terms in the syllogisms or
demonstrations of the mixed sciences, as mentioned in the preamble to
this paper.6

To explain this, it should be noted that physical qualities may be
divided into two types depending on their proximity to sense
experience. Some qualities are directly sensible, such as heat, color,
sound, odor, and taste, all of which can be sensed immediately by the
external organs of the body. Others are reductively sensible in that
they can be known only through sensible effects; examples of this type
are electricity, magnetism, and chemical affinities. Pertaining to this
latter division are also motive and resistive powers, such as gravity
and resistance to motion, which were already known to Aristotle.

Qualities in all these categories can be said to be quantified
simply because they are present in quantified bodies. Their quantity
can be measured in two ways, giving rise to two measures usually
associated with physical quality, namely, extensive and intensive
measurements. Qualities receive extensive quantification from the
extension of the body in which they are present; thus there is a
greater amount of heat in a large body than in a small body, assuming
both to be at the same temperature. They receive intensive
quantification from the degree of intensity of a particular quality in
the body. If two bodies are at different temperatures, for example,
there is a more intense heat in the body at the higher temperature, and
this regardless of the size of either. Generally the intensity of a
physical quality can be determined either from an effect, or from a
cause, or from a quantitative modality the quality produces in the
subject in which it is. An effect would be the change it produces in
another subject, a cause would be the agent that produces the intensity
in the subject, and a quantitative modality would be some concomitant
variation between the intensity of the quality and a quantitative
aspect of the subject. The subject bodies in such cases are called
instruments, and they can be of various types depending on the quality
being measured.

Through the use of instruments numbers can be assigned to
qualitative intensities and they can thus enter into mathematical
calculations. Along with the numbers, however, units of measurement
must usually be specified for qualities, just as they are for
quantities. For example, in the International System of Units now
standard in science, the unit for mass (m) is the kilogram, that for
length (l) is the meter, and that for time (t) is the second. Since
these are proposed as the primary dimensional units, units for
qualities are preferably expressed in terms of them. Such qualitative
dimensions are given in terms of the exponents or powers to which the
basic quantities (m, l, t) must be raised for their cumulative product
to express the proper dimensional unit of the quality being measured.
The only requirement put on these exponents is that they be numerical
constants; they may assume zero, negative, and fractional values. These
are shown in Table II for a few of the dimensional units that are
employed in work on mechanics and electromagnetism.

One of the simplest qualitative units in this table is that for heat
temperature (6), which can be measured through the expansion of mercury
in a thermometer, actually a measurement of length (l). Heat capacity
(7), on the other hand, measures the quantity of heat in a body. It has
the dimensionality of energy per unit mass per degree of temperature,
given in the table as m0lt-2, which reduces to
lt-2 since m0=1. Energy (5), or force times
distance or length, has the dimensional unit ml2t-2,
so when one divides this by mass (m) and length (l, the
dimensional unit for heat temperature), the resulting dimensional unit
is lt-2. Working in this way through the table, by applying
the standard definitions of the various terms it is possible to verify
the dimensional units that are there attached to them.

However, it turns out that not all qualitative measurements need
have a dimensional unit. This can be seen in the last three entries in
the table, those for specific heat (15), atomic weight (16), and
molecular weight (17). For all of these the exponents are zero,
indicating that this dimension is a pure ratio. Specific heat, by
definition, is the ratio of the heat capacity of a particular substance
to the heat capacity of water. Each heat capacity has the dimensional
unit explained above, but when the two are placed in simple ratio,
these units cancel out and the result is a pure number. The same thing
happens with atomic weight and molecular weight, as we are about to
explain. Both of these express ratios to a unit weight taken by
convention as one-twelfth the mass of the carbon atom (12C =
12.00000). Since the weight or mass units characteristic of various
atoms and molecules cancel out when placed in ratio to this unit
weight, their respective atomic and molecular weights likewise are pure
numerics.

Early Quantum Mechanics

With this we are in a position to return to our third "mixed
science," which we here present in its early form as it was first
presented by Niels Bohr and others. This made use of Planck's concept
of the quantum plus advances in chemistry and spectroscopic analysis,
as well as selected portions of classical mechanics and
electromagnetism, to present a unified view of elements and compounds
as known at the beginning of the twentieth century.

The four-element theory of Aristotle was, of course, the longest
lasting theory in the history of science, enduring from the fourth
century BC all the way to the early nineteenth century. The "hot-cold"
and "wet-dry" couplets of that theory prompted much work with furnaces
and solutions in alchemy, and also with Galenic medicine, up to the
Renaissance. By the nineteenth century metrical concepts were even
available for dealing with heat and fluidity, as can be seen in Table
II. But ultimately they proved inadequate for attacking the element
problem and had to give way to "gravity-levity" as the preferred
couplet for its solution.

In The Modeling of Nature we used the four forces of modern
physics to characterize the natures of inorganic substances, and at
this stage we introduce the first two of these, gravitational force and
electromagnetic force.7 With hindsight we can say that only
two basic instruments were required to investigate the effects of these
forces, the mass spectrograph for investigating gravitational forces,
and the spectroscope for studying the emission and absorption of
electromagnetic radiation. Advances in chemistry were also needed:
first, the discovery of various laws of chemical combination, leading
to demonstrations of the existence of atoms and molecules; second, the
systematic study of chemical combinations, leading to the periodic
table of the elements. Related to these were advances in physics: first
the discovery of the electron as the unit of electric charge; then of
the nucleus of the atom and its basic constituents, the proton and the
neutron. Some of the physico-mathematical demonstrations implicit in
these advances have been sketched in Modeling, and these may
be consulted for details.8

Bohr took all this information and synthesized it in a quantized
model of the atom in which electrons move around the nucleus in
planetary orbits, structured in various shells. Within these shells he
postulated that electrons move in stable orbits without emitting or
absorbing radiation, as they would in classical electromagnetic theory.
Under the influence of strong electrical fields or other external
energy, however, electrons can make stepwise jumps from one shell to
another. Bohr speculated that when an electron moves farther from the
nucleus in this way it absorbs an amount of electromagnetism determined
by the different energy levels of the two orbits; when it drops from an
outer orbit to an inner one, it emits a similar amount. By formulating
a series of rules stating which electron transitions are allowed and
which are not, Bohr found that he could explain the emission and
absorption spectra of many chemical elements. In effect, he could
correlate the wavelength and intensity of the radiation characteristic
of a particular element with the jumping of electrons in the atomic
model of that element from one orbit or energy state to another.

Further refinements of Bohr's model began with Arnold Sommerfeld's
replacement of circular orbits by elliptical orbits. Then, along with
that, was the possibility of the electron orbits having various
orientations in three-dimensional space, providing an azimuthal quantum
number, and its having different angular momenta, giving two more
energy states. Another was the introduction of electron spin, that is,
the rotation of an electron on its own axis, to make a fourth. In all,
therefore, each electron in an atom now had the possibility of existing
in four energy states, each denoted by a different quantum number. Yet
another was the introduction of a principle by Wolfgang Pauli
specifying that no two electrons in an atom could occupy the same
energy state at any one time, which was equivalent to stating that no
two electrons in an atom can have the same four quantum numbers. How
all these developments could be used to provide models of the hydrogen,
helium, and sodium atoms respectively has been explained in The
Modeling of Nature, from which my next illustration (Fig. 3) is
taken.9

These advances, it must be noted, were closely tied to studies of
the electromagnetic spectra of the different elements in the periodic
table. The four quantum numbers were based on calculations of
theoretical physicists using classical mechanics and electromagnetic
theory. By what seems a remarkable coincidence, counterparts of their
results could be found in four different types of spectral lines of the
elements identified by spectroscopists working in different portions of
the electromagnetic spectrum. These lines were described by them as
strong, principal, diffuse, and fundamental, and designated by four
letters (s, p, d, and f), which could be correlated with the four
quantum numbers of Bohr's theory.

To complete the picture provided by the Bohr-Sommerfeld atom it is
necessary to mention a device related to the spectroscope, namely, the
mass spectrograph, invented by the British physicist F. W. Aston in
1918. This instrument was designed to provide accurate measurements of
atomic weights. In it, ions or charged atoms are sorted out through the
use of electric and magnetic fields in such a way that their paths of
travel, and thus the positions at which they impinge on a screen or
photographic plate, provide a measure of their masses. Experiments with
the spectrograph show that the atoms of naturally occurring elements,
although occupying the same place in the periodic table (and hence
called "isotopes"), have nuclei of slightly different masses, depending
on the number of neutrons within them. The element chlorine, for
example, has an atomic number of 17; this specifies its number of
peripheral electrons, which must be balanced by 17 protons in its
nucleus, since the atom itself is electrically neutral. Normally
chlorine has an atomic weight of 35, which means that 18 neutrons must
be added to the 17 protons to give the proper weight. Some atoms of
chlorine, however, have an atomic weight of 37; in these, therefore,
there must be two more neutrons, or 20, to supply the additional
weight.

The most remarkable success of the Bohr-Sommerfeld model, however,
was the way in which it could explain all of the chemical properties of
the various elements in terms of the outermost electronic shells. These
accounted for all the known valencies of the chemical elements, and
thus gave a theoretical justification for the formulas used to describe
chemical changes. The model provided a remarkable heuristic for
expanding the science of chemistry, and brought it to the stage of
being the most developed of the "mixed sciences" in the twentieth
century.

Stable Natures: Chemical Elements and
Compounds

With this I return to Aristotle's concept of nature and take up the
second principle he identifies with nature, namely, natural or
substantial form. Natural form for him is nature, but it differs from
protomatter in that it is a real and actual principle, whereas
protomatter, though real, is only a potential principle. Of itself,
protomatter is unintelligible, but when actualized by form it becomes
intelligible in the substances we know through sense experience. The
case is different with natural form, for the human mind grasps it
directly and instinctively. Natural form provides the window through
which the world of nature is seen and through which many of the natures
inhabiting it can be readily understood.10

In the world of the living it is comparatively easy to grasp the
natural form of an oak, or a squirrel, or a horse, and so to answer the
question "What is it?" in a general way. From this we can proceed to
note what it has in common with, and is different from, other
organisms, and so formulate its scientific definition, say, in terms of
genus and species. Not so with inorganic substances. True enough, we
can know some elements in this way, say, carbon, sulphur, lead, and
gold. But can we know them as elements in the proper sense, that is, in
a way that enables us to differentiate them from, say, water, or salt,
or emerald? Before the development of chemistry as a science, and
particularly the knowledge provided in the periodic table, the answer
would have to be "No." Now, with the aid of the periodic table, it can
be "Yes." We can know the natures of inorganic substances, both
elements and compounds. This can be seen with the aid of the following
two figures.

The first (Fig. 4) is a novel way of presenting the periodic table
in a way that shows its relation to early quantum
mechanics.11 At the bottom center is shown the nucleus of an
element's atom, and rising above it are some ninety energy levels to
account for all the different states electrons can occupy in naturally
occurring elements ranging from hydrogen to radium and beyond. The
various shells are numbered in arabic numerals on the left, and these
correspond to the periods of the periodic table in roman numerals on
the right. Between the energy levels and the periods on the right are
the symbols for all the elements (except the rare earths).
Corresponding to these on the right are the divisions of shells into
subshells, and then the further divisions of these into the spectral
lines through which the energy levels of the different elements are
known. Note that these are given in terms of the four quantum numbers,
simply using the letters s, p, d, and f of spectroscopic analysis, and
saying nothing about the details of electron orbitals. Further shown
there is the order in which the various shells and subshells are filled
up with electrons, as one proceeds from the lightest element, hydrogen,
at the bottom, to the heaviest at the top.

My next transparency (Fig. 5) complements this peripheral electronic
structure of the atoms by showing in the upper display the components
of the nuclei and isotopes of the first ten elements in the periodic
table.12 The lower display exhibits the peripheral electrons
and valencies of the Bohr atom for the same elements, and below that
some examples of the two types of molecular bonding, ionic and
covalent, that result from these valency structures. These provide
examples of typical molecules: for ionic bonds, HCl, BeO,
NH3, and CaCl2; for covalent bonds,
H2, HCl, CCl4, and Cl2.

In The Modeling of Nature I made the case that the
information provided by the periodic table of the elements provides us
with a knowledge of their natures far superior to any comparable
knowledge we have of living organisms.13 All one need do is
consult the Handbook of Chemistry and Physics to find all the
essential features of the elements and their isotopes, plus their
properties, in consummate detail. Tables of their compounds in the same
source, both inorganic and organic, provide similar knowledge of their
natures. And all of this information is reducible to sense knowledge,
through the use of the metrical concepts we have explained above. What
is more, we need not base our knowledge on theoretical entities such as
elliptical orbits and spinning electrons. We tend to replace "element"
with "atom" and "compound" with "molecule," but it is the elements and
the compounds that fall directly under sense experience, not the atoms
and the molecules. And when we measure the spectral lines that reveal
energy levels, we do so in terms of wavelengths in the electromagnetic
spectra of elements we handle, spectra that are themselves visible,
either directly or reductively. The middle terms in our reasoning
processes are thus both physical and mathematical. So, to answer my
initial question, the mathematical physicist does have access
to the natures of the non-living, and he does so through sense
knowledge. Again, details will be found in The Modeling of
Nature.14

So, we know the natures of elements and compounds. Do we also know
the natures of planets and stars? I discuss this briefly in
Modeling and propose that the answer is "No," and this for the
simple reason that they do not have natures in the strict
sense.15 Of the heavenly bodies, some, such as planets and
asteroids, are mainly solids, whereas stars like our sun are
principally hot gases. Earth itself is a mixture or aggregate of many
different elements and compounds, held together by the force of
gravity. Similarly the sun is a mixture of hydrogen and helium in the
gaseous state. The unity of a star would seem to be analogous to the
unity of the earth: largely a mass of different substances held
together by natural forces of one type or another. And if current
models of planets and stars are correct, they can go through a process
of evolution and can have a history like many plants and animals. Yet,
unlike plants and animals, planets and stars have no natural form,
there is no unifying or specifying form within them guiding that
history toward some perfective state, as it does in the case of
organisms. The protomatter that is distributed throughout a star's
bulk, for example, is informed by a variety of elementary forms that
themselves are replaced by others, as the various potentials latent
within the substrate are actualized, until the mass-energy of the star
is exhausted and its ceases to exist as such. Much the same fate awaits
planets and asteroids, as they break up into their components and
dissolve into the elements and compounds that are their basic
components.

Later Quantum Mechanics: Waves and
Matrices

Let us turn now to later quantum mechanics, which has taken two
different forms, wave mechanics and matrix mechanics, both said to be
equivalent from the mathematician's point of view and both extremely
difficult to explain in non-mathematical terms. Wave mechanics, in
particular, raises interesting philosophical questions, possibly
because people can visualize wave packets and ponder how they can
travel at speeds faster than light, tunnel through barriers, perform
spooky actions at a distance, and in general do things offensive to
common sense. Related to this development are many discoveries in
high-energy physics, which has seen an enormous growth of so-called
elementary particles, subatomic wave-particles, beyond the three
mentioned thus far, electron, proton, and neutron.

But before discussing strange particles, we should make an important
point. Replacing orbiting electrons with wave functions or other
theoretical entities in no way calls into question what has already
been said. The observational basis for quantum mechanics, and the
information it provides of natures in the non-living, remains exactly
the same as previously: energy levels, revealed by spectral lines, and
the frequency of transitions from level to the other, revealed by the
intensity of spectral lines. The changes over the past fifty years are
in the way scientists theorize about what goes on in the interiors of
atoms and their nuclei, not about the experimental findings that ground
their theorizing.

Some idea of wave mechanics may be gained from the materials
displayed in my next transparency (Fig. 6).16 At the top is shown the famous wave
equation formulated by Erwin Schrödinger early in 1926. The
variable is the Greek letter psi, and so the equation is referred to as
the psi-function. What psi stands for, unfortunately, is the subject of
much dispute. Schrödinger originally thought it stood for electric
charge distribution within the atom, thus giving it a physical meaning,
but he later ruled this out as impossible. In 1929 Max Born gave psi a
statistical interpretation, saying that it represented the probability
of finding an electron at a particular place within the atom. This view
was vigorously rejected by Schrödinger. And finally in the early
1950's David Bohm gave psi a realist or deterministic interpretation,
holding that its results are determined by potentials within the atom.
So we have three views of the wave equation: an actual physical
function, a probability function, or a potentiality function. No
physicist currently holds the first view, but the second and third are
still the subject of vigorous dispute among physicists and philosophers
of science.

In actual practice the wave equation is most used by chemists, and
for their purposes it suffices to interpret the equation statistically
as a probability function. The wave equation for the hydrogen atom can
be solved mathematically, and among its solutions are the six functions
graphed in the upper left quadrant of the transparency. The ordinate
here is probability. The functions are referred to as radial density
plots of hydrogen orbitals, and they are labeled from (a) to (f) as the
1s, 2s, 3s, 2p, 3p, and 3d. The important thing to note here is the way
in which the orbitals are designated -- in terms of shells (1, 2, 3)
and subshells (s, p, d). This is spectroscopic terminology, not the
terminology of the Bohr atom. In addition to radial density plots,
moreover, it is possible to calculate the boundary contours of the
orbitals, to give some idea of their orientation in three-dimensional
space. The results for several of these are shown in the series of
figures on the right side of the transparency, with the s orbital on
the top directly under the wave equation, the three 2p orbitals under
that, and the five 3d orbitals at the bottom. Some relationship can be
discerned between these diagrams and the elliptical orbits of the
Bohr-Sommerfeld atom, but the differences are more marked than the
similarities.

The final diagram is that on the bottom left quadrant, which
illustrates how chemists use the wave equation to explain the bonding
of atoms within a molecule. The molecule here is beryllium chloride,
and the atoms are one beryllium atom and two chlorine atoms. The (a)
part of the diagram shows the two chlorine atoms with their orbitals on
either side of the beryllium atom with its orbitals before their being
joined. The (b) part shows how actual bonding occurs, with the
BeCl2 molecular orbitals now serving to explain the binding
forces between the component atoms. It seems obvious that calculations
of chemical bonds using wave mechanics yield results that are far
superior to the much simpler insights provided by the older quantum
mechanics of Bohr and Sommerfeld.

Wave mechanics thus is a potent instrument for studying the
functions of electrons within atomic and molecular structures. It can
also be used to speculate about electrons and photons when they are
conceived as individual wave pulses outside the atom. Here the
mathematics becomes complex, partly because of the infinities involved.
The problem may perhaps be understood by a simpler example that has its
roots in Aristotle. In the Posterior Analytics I,13 Aristotle
observes that "it belongs to the physician to know that circular wounds
heal more slowly [than other kinds], but it belongs to the geometer to
know the reasoned fact" (79a15-16). The reason is that healing occurs
along the perimeter of a wound, and the circle has the smallest
perimeter for any given area; thus it will heal more slowly. Using
calculus, one can also calculate the rate of healing. What is involved
is a function which starts with the wound's area, A, and decreases
exponentially with the passage of time. The fit is remarkable for all
points along the curve except at the end. The mathematical function
approaches the x-axis asymptotically, which means that it never reaches
the x-axis, or, as some say, it meets the x-axis at infinity. But here
nature doesn't obey the mathematics. At some point in time nature
closes the wound and A goes to zero.

In mathematical physics, as noted at the beginning of this paper,
two premises are ultimately involved in any proof, one a physical
premise, the other a mathematical premise. Which premise should take
priority in case of conflict? In my view the physical premise must be
regulative over the mathematical. This goes contrary to much
contemporary discussion of quantum mechanics, where, it seems to me,
mathematics is driving the arguments. If physics is not, then the
philosophy of nature has little or nothing to contribute to the debate,
which perhaps explains why it is consistently ignored.

A final word about matrix mechanics. The basic mathematics behind
matrix mechanics is the same as that behind wave mechanics, but it uses
a different formulation, one proposed by Werner Heisenberg in 1925.
Heisenberg became concerned about Schrödinger's wave equation
because psi was not an observable, and he thought physics should stick
to observables. Accordingly, he saw the goal of quantum mechanics to be
the computation of two matrices, one a diagonal matrix which would list
the observed energy levels of atoms and molecules, the other a related
matrix that would list the transition probabilities between the various
levels. In both cases one would be concerned with observables or
measurements: the wavelengths of spectral lines and their intensities,
both of which are available to the physicist. Heisenberg also saw these
matrices as grounded in the potentia of Aristotle, his term
for protomatter. As I see it, his view of the psi-function was
ultimately a potentiality function, although he is often listed as
following the Copenhagen interpretation, which sees it as merely a
probability function.

Strange Particles

Our exposition thus far has been based on two powers distinctive of
inorganic natures, gravitational force and electromagnetic force. At
this point we must consider the two remaining powers, the strong force
and the weak force, for knowledge of these is the main fruit of
research in high-energy physics. To put these powers in context, Fig. 7
shows the main models we have been using thus far: at the top, the
basic composition of the natural body, with its powers and sensible
qualities on the right; in the middle, the Aristotelian model, with the
basic couplets, hot-cold, wet-dry, and heavy-light, all directly
perceived by the senses; and at the bottom, the modern analogue with
the four forces of modern physics. Those on the left, gravitational
force, explains mechanical motions, and electromagnetic force, the key
concept, along with mechanics, explains the periodic table. Now we turn
to the strong force, invoked to explain nuclear reactions, omitting,
for lack of time, the weak force, which is required to explain
radioactivity.17

The strong force is best explained in terms of the quark hypothesis,
introduced in 1964 by Murray Gell-Man but preceded in 1961 by his
"Eightfold Way," which can serve to illuminate his later work.
Analyzing the results of many experiments in high-energy physics,
Gell-Man arrived at a number of qualitative dimensions that seem to be
conserved in strong interactions.18 Among these four are
important for our purposes, as shown in my next illustration (Fig. 8).
They are: atomic weight or mass (A) and electric charge (Q), both of
which we have already mentioned, and two other dimensions that are new,
hypercharge (Y) and isotopic spin (I). Measuring values of these for
all the meson and baryon states then known, Gell-Mann observed the
symmetries displayed in the octet diagrams below Table III, that in the
middle for mesons, whose atomic mass number is zero, that at the bottom
for baryons, whose atomic mass number is one. Isotopic spin can be
thought of as the spin of the nucleus of an isotope of an element, and
on that basis may be intelligible. More mysterious is hypercharge,
which Gell-Mann thought was something like a strong charge, although it
produced strange V-shaped effects in cloud chambers. In view of these
effects, he later changed the dimension to "strangeness."

Let us compare Gell-Mann's eight-fold way to the simple qualitative
matrix we used earlier to characterize the transformation of the
elements in Aristotelian physics. For Aristotle, changes between the
elements are possible only when one primary quality is conserved and
the other modified. For Gell-Mann, this is likewise the case, for all
particle reactions are based on some qualitative dimension being
conserved and the other being modified. The abscissae on the diagrams
represent electric charge (Q), while the ordinates are given for both
isotopic spin (I) and hypercharge (Y). Notice that a diagonal
transition is permitted in Gell-Mann's system, whereas it is not in the
Aristotelian case. This seems to be possible because the mass number is
conserved for either mesons or baryons, while both Q and I (or Y) are
changing. In Aristotle's system, where only two parameters are variable
(H and W) and there is no constant such as mass number to be conserved,
diagonal transitions are automatically disallowed. But there is no
reason to reject the possibility that protomatter lies behind both
matrices as the ultimate conservation principle in the ontological
order.

One could well question the sense in which "strangeness" can be a
qualitative measure of an elementary particle. Apparently encouraged by
his introduction of this mysterious attribute, Gell-Mann went on to
formulate his quark hypothesis, which added a new entity, the quark, as
the fundamental component of both mesons or baryons.19 The
quark has many possible attributes, and when these are present in
combination there can be thirty-six kinds of quark. According to the
Standard Model, quarks come in different "flavors" and "colors," as
explained on my next transparency (Fig. 9). As to flavors, there are
six to choose from: up, down, strange, charm, bottom, and top. For each
combination of these attributes there is an anti-particle with the same
mass but opposite charge. As to colors, the choices are red, green, and
blue. The names obviously do not mean anything: the pairs could be
hot-cold as easily as up-down, or wet-dry as easily as bottom-top. All are
related in one way or another to indirect measurements performed on
nuclear components when these are subjected to high-energy cosmic rays
or to heavy bombardment in particle accelerators.

Of all the possibilities, it is noteworthy that only two quarks, the
up and down quarks, are stable; all the others enjoy but a transitory
existence. Moreover, the general rule is that all baryons are composed
of three quarks and all mesons are composed of a quark and an
anti-quark. No one has ever observed a single quark, and so they are
presumed, like protomatter, to be incapable of existing by themselves.
They are also thought of as permanently "trapped" or confined within
the baryons and mesons of which they are parts. And as to the baryons,
only two of these are stable particles, the proton (composed of an
up-quark, its anti-particle, and a down-quark) and the neutron (composed
of a down-quark, its anti-particle, and an up-quark). All others
likewise have a fleeting existence, probably representing transient
states of matter at different stages in the formation of the
universe.

The result of all this investigation, extending over fifty years, is
that we are still left with only three basic particles, the proton, the
neutron, and the electron. And none of these can be said to have a
nature in the strict sense, although they enter into the composition of
the elements and compounds whose natures we do know, as already
explained. High-energy physicists provide evidence for the existence of
mesons and baryons by the hundreds, of bosons such as the photon and
gluons of different types, of leptons such as the electron, the
positron, and various kinds of neutrinos. All of these are only
transient entities, entia vialia, to use the Latin
expression.20 We may think of them as transient natures, but
this is not the sense in which Aristotle intended the term "nature." On
the other hand, he did intend nature to mean protomatter. And here we
have the clue to the significance of particle physics. Protomatter is
the ultimate substrate, the basic potency that underlies the operations
of nature. The different groupings of strange properties fail to yield
a particle ultimate, but Aristotle's ultimate is still there, the
potential correlate of the natures we have come to know in the world of
the inorganic.

NOTES

1. The full title of the book is The Modeling of Nature:
Philosophy of Science and Philosophy of Nature in Synthesis
(Washington, D.C.: The Catholic University of America Press, 1996). The
paper is entitled "Thomistic Reflections on The Modeling of
Nature: Science, Philosophy, and Theology," forthcoming.

3. The materials treated here are summarized from my "Elementarity
and Reality in Particle Physics," Boston Studies in the Philosophy
of Science 3 (1968): 236-271, reprinted in my From a Realist
Point of View: Essays in the Philosophy of Science, 2d ed.
(Lanham-New York-London: University Press of America, 1983), 185-212.

4. See The Modeling of Nature, 58-63, explained more fully
in "Thomistic Reflections."

5. Mathematical models that might help one understand this
development were omitted from The Modeling of Nature because
of their complexity. They were introduced in Part II of "Thomistic
Reflections" and are further developed in what follows.

6. On metrical concepts, see The Modeling of Nature,
239-244 and 409-414; also my "The Measurement and Definition of Sensible
Qualities," The New Scholasticism 39 (1965): 1-25, reprinted
in From a Realist Point of View, 2d ed., 73-97.

17. For modeling the inorganic, see The Modeling of Nature,
53-58 and 70-73.

18. Again see my "Elementarity and Reality in Particle Physics"
(note 3 above). Further information is given in G. D. Chew, M.
Gell-Mann, and A. H. Rosenfeld, "Strongly Interacting Particles,"
Scientific American 210 (1964): 74-93. For technical details
see M. Gell-Mann and Y. Ne'eman, The Eightfold Way (New York
and Amsterdam, 1964).

19. The basic source here is Murray Gell-Mann, The Quark and the
Jaguar: Adventures in the Simple and the Complex (New York: W. H.
Freeman and Company, 1994).